Experiments have demonstrated that the mechanical stretching of bulk polyethylene can increase its thermal conductivity by more than two orders of magnitude, from 0.35 W/mK to over 40W/mK, which is comparable to steel. This strong effect is believed to arise from the increased alignment of the constituent polymer chains, which are thought to have very high thermal conductivity. Although it is well established that bulk polymers have low thermal conductivity, these experiments suggest that cheap, high thermal conductivity polymer materials can be engineered. This type of advancement may provide a much cheaper alternative to the conventional metal-based heat transfer materials that are used today. In order to quantify upper limits on the thermal conductivity of polyethylene, we examine the underlying phonon (lattice wave) transport using molecular dynamics simulations. We first show that the thermal conductivity of individual polyethylene chains is high, and can actually diverge (approach infinity) in some cases. We then discuss how the high thermal conductivity of individual chains is reduced by the presence of additional chains, through van der Waals chain-chain interactions. These intermolecular interactions give rise to both a 2D planar lattice structure and a 3D bulk lattice structure, which allows for the observation of an interesting 1D-to-3D transition in phonon transport.(cont.) For most crystalline nanostructures, the thermal conductivity decreases with decreasing crystal size from an enhanced boundary scattering of phonons. In the case of polyethylene, however, the intermolecular chain-chain interactions increase phonon-phonon scattering along each chain and actually result in the opposite trend, where the thermal conductivity increases with decreasing crystal size. The results provide important fundamental insight into phonon-phonon interactions and will also aid in the design and structural optimization of high thermal conductivity polymers.